We used resonant inelastic x-ray scattering to reveal the nature of magnetic interactions in Sr ₂IrO ₄, a 5d transition-metal oxide with a spin-orbit entangled ground state and J _eff=1/2 magnetic moments. The magnon dispersion in Sr ₂IrO ₄ is well-described by an antiferromagnetic Heisenberg model with an effective spin one-half on a square lattice, which renders the low-energy effective physics of Sr ₂IrO ₄ much akin to that in superconducting cuprates. This point is further supported by the observation of exciton modes in Sr ₂IrO ₄, whose dispersion is strongly renormalized by magnons, which can be understood by analogy to hole propagation in the background of antiferromagnetically ordered spins in the cuprates.